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Four times the square of a number is 45 more than eight times the number. What is the number?

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3 Answers

The answer is tricky but it is -5/2 & 9/2. 

This is an algebraic formula in the form of words.  It looks like this, 4x2 = 45 + 8x.   I will breakdown how I got to the formula below.

1. four times a number means  4 times x because we do not know what "the number" is (4x)

2. the unknown number or 'x' is squared because the question says, "the square of a number" (4x2)

3. the word 'is' typically indicates "equal to" (4x2 =)

4. the 45 given (4x2 = 45)

5. the words, 'more than' typically mean to add to something so that is where the addition comes from (4x2 = 45 +)

6. "8 times the number" refers to 8 times the same number that was spoken of in the first part of the question  (4x2 = 45 + 8x)

7. Now that we have the equation, we can solve for x.

Let x be the unknown number. Then fout times the square is 4 x^2. Eigth times the number is 8x. The difference between them is 45, 4x^2 - 8x = 45. Add up 4 to both sides of the equation. You will obtain

                                        4x^2 - 8x +4 = 49     or  4(x-1)^2 = 7^2

Take square root of both sides to get two eqautions (you have two solutions because the eqaution is quadratic):     2(x-1)  = 7    and        2(x-1) = -7

or                 2x -2 = 7              and 2x -2 = -7

We have       2x = 9 or   x = 4.5               and 2x = -5 or     x = -2.5.  Conditions of the problem hold:

     4 (4.5)^2 - 8 (4.5) = 81-36 = 45       4(-2.5)^2 - 8(-2.5) = 25 +20 = 45

4x^2 = 8x+45

Collect all terms in one side,

4x^2 - 8x - 45 = 0

Factor,

(2x+5)(2x-9) = 0

Answer: x = -5/2, 9/2